Abstract A new method for the computation of a reduced order circuit model, based on numerical elimination of the internal circuit nodes, is presented. The circuit passivity is preserved unlike the majority of other approaches. A 1033 node RC circuit is reduced to an admittance with one pole and two zeros for a frequency range of three decades.

1. INTRODUCTION In the last years the reduced order circuit models have been proved their usefulness to the fast simulation of electromagnetic field models in integrated circuits. These models are describing interconnection structures as well as passive components. Some specific methods have been developed in the literature: Asymptotic waveform evaluation (AWE) method [1], Complex frequency hopping (CFH) method [2], Padé via Lanczos (PVL) method [3], and PRIMA [4]. A very important feature of this kind of methods is passivity preservation. An order reduction method has this property if any reduced model corresponding to a passive original (high order) circuit is passive. Only one [4] of the known reduced order methods preserves passivity. The aim of this paper is to analyze the efficiency of the internal node elimination [5,6] as a tool of obtaining reduced order models. Both the symbolic and numerical elimination procedures are considered. A very simple reduced order model of a large scale circuit example is found. 2. INTERNAL NODE ELIMINATION Writing the equations of a numerical method (finite difference method or finite element method) for solving an electromagnetic field problem in which the inductive effect is negligible is equivalent to the building of a regular structure circuit with branches having a simple structure (for example a parallel RC circuit). The parameter values of these elements depend on the material properties and on the discretization step. Consider a circuit of this kind in Fig.1 having the nodal admittance matrix Y. The circuit equations are YV = J where the entries of V are node voltages and the entries of Y are yij = ±Gij ± s × Cij . It follows that the diagonal

Fig. 1 entries of Y are RC admittances as well as the non-diagonal ones with the minus sign.

the internal node elimination in a regular structure RC circuit used for the electromagnetic field computation doesn’t necessarily lead to a circuit made only of RC admittances. in a regular structure circuit · · For The
y ij new i= j y = y
and a double zero occurs if
y ij
old
= 0.
a similar statement can be proved. a difference of
RC
two RC admittances. and V2 ' any circuit equation except the first four can be used. The transformation of an intricate RC admittance into a simpler one may be the basis of a passivity preserving algorithm. if
i¹ j
.s × C ij .
y ij
new
is not necessarily an RC admittance
RC (1) RC ( 2) RC
because the simple pole/zero alternation on the real axis may not be ensured (for example. in the best case.ATEE-2004
A function of the complex variable s is an RC admittance if and only if the following conditions are fulfilled: · is a rational fraction of s with real coefficients. y iq = -G . The following remarks on the node elimination algorithm can lead to some useful conclusions: · The
y ij new
computed using a diagonal pivot . V2 . Even though the internal node elimination doesn’t lead to a symbolic method for the computation of a reduced order model valid in a certain frequency range. the closest to the origin being a zero. y ij iq iq
so that
y ij
new
= y ij
old
y -
(1) RC y
×y
RC (3)
. V p can be eliminated using the equation q in which we assume that yq p ¹ 0 . the expression of the yij new numerator and denominator become very intricate in a large scale circuit. · the poles and zeros are simple and alternate on the negative real axis.
case.K yq n yq p × Vn
This is the well known Gaussian elimination and it is used only for the entries for which y q j × y i p ¹ 0 [5.
y qj = -G ( 2)
qj
. This is because a simplification procedure is mandatory during the node elimination process.
y qq = G qq + s × C qq
y q p (q = p)
can be an RC admittance. To conclude.
new
y ij
computed
y (1) RC -y ×y RC (3)
using
( 2)
an
off-diagonal
y ij new
pivot
y q p (q ¹ p)
leads
to
= y ij
old
so that
may be.s × C . it may be used in
. It follows: V p = and each entry of Y will be updated as:
yij new = yij old y q j × yi p yq p
yq 1 yq p × V1 yq 2 yq p × V2 . V1 ' .6]. In order to eliminate all unknowns except V1 . in this
old = -G ij . The possibility to build a procedure of this type is analyzed in the following.s × C qj . For example. · the number of zeros is equal or greater with one with respect to the number of poles. If the node elimination is performed repeatedly using the complex Laplace variable s as a symbol.

y 11 22 21
. A procedure giving the reduced order circuit having a given Y RC (w ) within a given error margin ε w. the symbolic techniques may be more efficient for repetitive computations than the numerical ones if the repetition number exceeds a certain limit. 2 The procedure for finding the reduced order circuit can be outlined as follows: · solve the field problem for a set of frequencies both for u1 = 1 and
u1 = 0 and u 2 = 1
u2 = 0
and for
and compute
y . The circuits we are dealing with have smooth frequency characteristics. Considering u1 = 1 and u 2 = 0 we compute y11 = i1 and y 21 = i 2 . y . that this is a passive RC admittance YRC . As it is well known [8].
where f(s) is a simple fraction corresponding to a
pole)
. In this case the numerical methods have a better efficiency. For a two port device the parameters of the voltage controlled representation can be computed by solving two analysis problems: 1. This algorithm preserves the passivity of the original RC circuit.
· · · ·
identify the poles of the reduced models of y11 and of y 22 using the method described in the next section with a given ε choose the circuit pole set as a minimal set satisfying the error margin ε for both
y11 and of y 22
choose the residue set of y 21 so that the condition k11 × k 22 . the original one is described in the next section. For an one port device the internal node elimination leads to an admittance connected between the port nodes. k . each circuit corresponding to a common pole of y11 . and y 21 .2. k for each two port corresponding to a pole
a b c
(y
a
=k f s . and y 21 = y12 the Cauer method of synthesis [7] can be used for finding the reduced order model. We can compute either the symbolic expression of Y RC (w ) or some values (for certain ω’s) of it.ATEE-2004
numerical computation.r. 2. y =k f s a b b c c
()
()
( ). using the energy functions [7]. as minimum multiplication or minimum fill. y 22 . Two outstanding properties related to this synthesis method must be taken into account: · the poles of y 21 = y12 must be poles of y11 and of y 22 · this method leads to a parallel connection of two ports in Fig.
Fig. y 22 . As a matter of fact some pivot selection strategies can be developed. the frequency range of interest being up to five decades. To this end some local optimality criteria.t. Considering u1 = 0 and u 2 = 1 we compute y 22 = i 2 and y12 = i1 . It can be proved. y =k f s .in criterion may be used.k 2 21 £ 0 be satisfied for each pole compute the values n. each one corresponding to a sparse matrix solving algorithm. Knowing y11 . k .

o. synthesis of the reduced order circuit The frequency range of interest is [ω m. the passivity of the original circuit being preserved. The first pole p1 is placed to the last value before that corresponding to an error of 2ε or greater. The complexity of the reduced model depends on the relative error ε with respect to the original characteristic. Synthesis is performed using a Foster. REDUCED ORDER MODEL OF YRC The poles and zeros of a RC admittance YRC(s) alternate on the negative real axis. 4. ωM].s.ATEE-2004
3. computation of the first zero z1 2.3). a Cauer or a Foster-Cauer procedure. The shape of the curve |YRC (jω)| vs. Characteristic
Fig. the closest to the origin being a zero. ω is defined by the location of the poles and zeros. Sweeping the ω axis starting from the origin we remark that the occurence of a zero is associated with a slope change of 20 dB decade and the occurence of a pole is associated with a slope change of . Obviously a greater ε leads to a simpler reduced model. The first asymptote is translated so that a maximum error of ε is obtained. the error between the asymptote of 20 dB decade and the given characteristic is checked. 20 dB decade . This is because the characteristic |YRC (jω)| has asymptotes whose slopes are 20 dB decade . Sweeping the frequency axis with a step ∆ωm. By this way a simpler passive RC admittance is obtained. EXAMPLE The case of a regular structure RC circuit resulting from a field problem in a capacitor (Fig. 4) is analysed
. 3 approximation by asymptotes has the maximum error of 3dB at the asymptote intersection (Fig. z1 is set to ωm. If this error occurs after the first step ∆ωm then p1 is placed very close to z1 . 0. Our algorithm for finding the reduced order model of a RC circuit has the following steps: 1.20 dB decade [9]. computation of the other poles and zeros 3. A natural way to approximate |YRC (jω )| is to consider fewer asymptotes with the same slopes as the original ones. The other asymptotes are determined without translation using the condition error ε in each frequency interval corresponding to the given asymptote. 0 a.

out. 6 is obtained. and 2801 capacitors.664661973608175641 The frequency characteristic | YRC (jω )| of the original circuit and of the reduced model are given in Fig.1263218120909251 C 2 = 0. 7. 2801 resistors.
5. Due to the smooth nature of | YRC (jω )| only 15 frequencies are enough.
Fig. The input current values are read from the PSPICE file *. The frequency range of interest is 100MHz . A program written in the MAPLE language is used to call PSPICE and to compute the reduced model. The symbolic elimination produces intricate expressions for which a simplification procedure which preserves passivity has not been found.5dB a reduced model with two zeros and one pole is obtained. Setting ε=1. The very good agreement between these characteristics is obviously. 6 The parameter values given by MAPLE are:
C1 = 0. 5
A discretization network of 26x8x5 points (Fig.
. The large scale passive RC circuits arising in the numerical electromagnetic field computation are taken into account.100 GHz. 4
Fig.21256279859459353022 × 10 -13 R1 = 1524.CONCLUSIONS The symbolic and numerical internal node elimination is analyzed in order to be used for building the reduced order circuit models. Using the Cauer I synthesis the circuit in Fig.50395853046378024786 × 10 . so the numerical AC analysis of PSPICE is used to obtain the frequency characteristic.7 R2 = 31.5) leads to a circuit with 1033 nodes.ATEE-2004
.
Fig.